{
  "id": "0811.0791",
  "version": "v2",
  "published": "2008-11-05T18:38:39.000Z",
  "updated": "2009-06-05T00:21:55.000Z",
  "title": "The Hilbert Transform of a Measure",
  "authors": [
    "Alexei Poltoratski",
    "Barry Simon",
    "Maxim Zinchenko"
  ],
  "comment": "18 pages",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $\\fre$ be a homogeneous subset of $\\bbR$ in the sense of Carleson. Let $\\mu$ be a finite positive measure on $\\bbR$ and $H_\\mu(x)$ its Hilbert transform. We prove that if $\\lim_{t\\to\\infty} t \\abs{\\fre\\cap\\{x\\mid\\abs{H_\\mu(x)}>t\\}}=0$, then $\\mu_s(\\fre)=0$, where $\\mu_\\s$ is the singular part of $\\mu$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-06-05T00:21:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42A50",
      "26A30",
      "42B25"
    ],
    "keywords": [
      "hilbert transform",
      "finite positive measure",
      "singular part"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0811.0791P"
    }
  }
}